Sigma-local graphs

Bose, Prosenjit; Collette, Sébastien; Langerman, Stefan; Maheshwari, Anil; Morin, Pat; Smid, Michiel

doi:10.1016/j.jda.2008.10.002

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Neighborhood or proximity graphs create a geometric structure that connects two points if they are close in some sense. These graphs have been well studied and include the relative neighborhood graph [16], the Gabriel graph [14], β -skeletons [18], σ -local graphs [2] and Delaunay triangulations [12]. A subset of these, called the empty region graphs, define a neighborhood graph where two points are connected if a geometric region parameterized by those points does not contain any other point [3].…”

Section: Related Workmentioning

confidence: 99%

“…Spectral clustering has been applied successfully in a number of fields, including image segmentation, text mining, and data analysis in general. However, there remain a number of open questions: (1) How to define the neighborhood around data points to estimate a "good" affinity matrix, (2) how to adapt the algorithm to account for variations in local scale or density of the data, and (3) how to automatically select the number of clusters. This paper concerns the former two questions.…”

Section: Introductionmentioning

confidence: 99%

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Locally-scaled spectral clustering using empty region graphs

Correa

Lindström

2012

Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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This paper introduces a new method for estimating the local neighborhood and scale of data points to improve the robustness of spectral clustering algorithms. We employ a subset of empty region graphs -the β -skeleton -and non-linear diffusion to define a locallyadapted affinity matrix, which, as we demonstrate, provides higher quality clustering than conventional approaches based on k nearest neighbors or global scale parameters. Moreover, we show that the clustering quality is far less sensitive to the choice of β and other algorithm parameters, and to transformations such as geometric distortion and random perturbation. We summarize the results of an empirical study that applies our method to a number of 2D synthetic data sets, consisting of clusters of arbitrary shape and scale, and to real multi-dimensional classification examples from benchmarks, including image segmentation.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Locally-scaled spectral clustering using empty region graphs

Correa

Lindström

2012

Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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“…Neighborhood or proximity graphs create a geometric structure that connects two points if they are close in some sense. These graphs have been well studied and include the relative neighborhood graph [30], the Gabriel graph [24], β -skeletons [33], σ -local graphs [5], Θ-graphs [32], γ-neighborhood graphs [54] and Delaunay triangulations [22]. These graphs have been well studied in terms of their geometric properties [4,8,14], and have been applied in geographic analysis [34], pattern recognition [50], clustering [52], machine learning [49], normal estimation [41] and the extraction of contour trees [39].…”

Section: Related Workmentioning

confidence: 99%

Towards Robust Topology of Sparsely Sampled Data

Correa

Lindström

2011

IEEE Trans. Visual. Comput. Graphics

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Topology of a sparsely sampled 2D terrain for 700 (top) and 4,000 (bottom) random points. Neighborhoods associated with the k-nearest neighbors and the Gabriel graph often introduce false extrema (red). A denser variant, the diamond graph, considerably reduces the number of false extrema, while our relaxed empty region graph accurately extracts the correct extrema, requiring only a marginal number of additional edges per data point over the Gabriel graph.Abstract-Sparse, irregular sampling is becoming a necessity for reconstructing large and high-dimensional signals. However, the analysis of this type of data remains a challenge. One issue is the robust selection of neighborhoods -a crucial part of analytic tools such as topological decomposition, clustering and gradient estimation. When extracting the topology of sparsely sampled data, common neighborhood strategies such as k-nearest neighbors may lead to inaccurate results, either due to missing neighborhood connections, which introduce false extrema, or due to spurious connections, which conceal true extrema. Other neighborhoods, such as the Delaunay triangulation, are costly to compute and store even in relatively low dimensions. In this paper, we address these issues. We present two new types of neighborhood graphs: a variation on and a generalization of empty region graphs, which considerably improve the robustness of neighborhood-based analysis tools, such as topological decomposition. Our findings suggest that these neighborhood graphs lead to more accurate topological representations of low-and high-dimensional data sets at relatively low cost, both in terms of storage and computation time. We describe the implications of our work in the analysis and visualization of scalar functions, and provide general strategies for computing and applying our neighborhood graphs towards robust data analysis.

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